How to add more geometry via animation?
It's very easy to animate the growth of the pipes if they were simple curves with a bevel shape. You would then animate the Bevel Factor End value to have the shape grow. This would not give you the ball joints you show in the image. You would have to treat them separately.

Another option would be to create the pipes as mesh objects and then use the Mask Modifier in combination with a Dynamic Paint object to reveal the mesh over time. This would give you more flexibility on the shape of the pipe you want to use.

To do this you would:


Create a pipe mesh object.
Create a mesh object to paint with (like a sphere) and position it at the beginning of the pipe.
Animate the brush object on a path that travels from the beginning of the pipe to the end.
In Physics tab, add Dynamic paint to the pipe object, set it as a Canvas and add new canvas. Set the Surface Type to Weight. You'll see a name appear in red in the Dynamic Paint output Vertex group box. Click the plus sign beside it to activate it as a vertex group for the object.
Add Dynamic paint to the brush object. Choose Brush as the type, add brush. 
Play the animation, you should see the weight colour of the pipe change as the brush object travels across it.
You can then add a Mask Modifier to the pipe object and choose the vertex group you created in the DynPaint panel.
You will probably need to add SubDiv mod to the pipe to create more vertices to play with so the reveal transition is smoother.